Discrete Geometry and Symmetry (15w5019)

Arriving in Banff, Alberta Sunday, February 8 and departing Friday February 13, 2015


(University of Calgary)

(Northeastern University)

(York University)


Discrete geometry (in a broad sense) investigates discrete structures in geometry and combinatorics such as polytopes, polyhedra and maps, tessellations (tilings), complexes and graphs, efficient sphere arrangements, packing and covering arrangements, and lattices. In focusing on these topics, our main goal is to concentrate on aspects of symmetry in the analysis and classification of these structures, as well as nourish new developments and explore unexpected connections. The themes of the proposed workshop are reflecting the rich mathematical traditions of the groundbreaking works of H.S.M. Coxeter, Laszlo Fejes Toth and Branko Grunbaum to whom we owe a great deal of our present understanding of discrete geometry, as well as the more recent progress that they inspired.The last three decades have seen a revival of interest in discrete geometric structures and their symmetry. In the area of polytope-like structures and symmetry, much of the recent progress has centred around the modern theory of abstract polytopes and combinatorial symmetry [8,9]. Abstract polytopes are combinatorial structures with distinctive geometric, algebraic, or topological properties, in many ways more fascinating than traditional polyhedra, polytopes and tessellations. While much of the fundamental work of Coxeter and Grunbaum in this area focused on highly regular structures, recent research also dealt with somewhat less restricted aspects of symmetry, thus broadening the classical approach while leading to many new and unexplored problems. Particularly noteworthy is the recent breakthrough result by Pellicer [11] on the existence of chiral polytopes of any rank (see also [3]). The rapid development of abstract polytope theory has resulted in a rich theory featuring an attractive interplay of methods and tools from discrete geometry (classical polytope theory), group theory and geometry (Coxeter groups and their quotients, as well as reflection and crystallographic groups), combinatorial group theory (generators and relations), and hyperbolic geometry and topology (tessellations and their groups); see [2,12] for two additional examples. Still, even after an active period of research, many deep problems have remained open and await solution. Researchers from diverse backgrounds will gather to build on earlier success and explore connections between the various discrete structures with symmetry as the unifying theme. The 50th anniversary in 2014 of the publication of Fejes Toth's classic on "Regular Figures" by Pergamon Press in 1964, as well as the upcoming Fejes Toth Centennial celebrations in Budapest in 2015, make the proposed Workshop a particularly timely event.Kepler was the first to formulate the discrete geometric problem of finding efficient packings of spheres (balls) in 3-space, and his conjecture about the most efficient arrangement became known as the Kepler Conjecture (finally confirmed by Hales in [6]). The systematic research on general packing and covering problems in space began in the late 1940's with the pioneering work of Laszlo Fejes Toth. The Hungarian geometry school of Fejes Toth greatly contributed to the growing field of discrete geometry and has attracted the interest of numerous other mathematicians, including prominent researchers such as Coxeter, Rogers, Penrose, and Conway. Two particularly active subject areas, already highlighted in Hilbert's 18th Problem, stand out since the early days of discrete geometry and are naturally intertwined, namely dense sphere packings, and tiling theory. The interest in sphere packings generated a great deal of research on the geometry of Voronoi tilings with a staggering number of real world applications. Over the past few years, it has become increasingly evident that further progress on most "hard problems" about efficient arrangements of solids would also require new optimization techniques, as emphasized in the recent books [1] and [7]. Phrased differently, while many central and by now classical problems in discrete geometry have an established record of strong connections with geometric analysis, coding theory, group theory (symmetry groups), number theory, and differential and integral geometry, the connections with combinatorics and optimization are of particular importance and have not yet been fully exploited. Advanced optimization techniques have significantly helped achieve recent breakthrough results on kissing numbers and densest lattice sphere packings [4,10].The purpose of the proposed workshop is to bring together established experts and junior researchers to share recent developments and emerging directions, encourage interaction and new collaborations, and achieve further progress on pressing major problems in the field. We are planning to schedule a number of key lectures by international experts, surveying the state-of-the-art and addressing existing connections. Other participants will have the opportunity to propose 25-minute talks to present their work, but due to the time constraints only a relatively small number of talks will be accepted. Our workshop schedule would provide ample time and opportunity for participants to interact and engage in mathematical discussion. References[1] K. Bezdek, Lectures on Sphere Arrangements - the Discrete Geometric Side, Fields Institute Monographs, Vol. 32, Springer, New York, 2013.[2] M. Conder, The smallest regular polytopes of given rank, Advances in Mathematics 236 (2013), 92-110.[3] M. Conder, I. Hubard and T. Pisanski, Constructions for chiral polytopes, Journal London Math. Society 77 (2008), 115-129.[4] H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Annals of Mathematics 170/3 (2009), 1003-1050.[5] J.H. Conway, H. Burgiel and C. Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., Wellesley, MA, 2008.[6] T.C. Hales, A proof of the Kepler conjecture, Annals of Mathematics 162/3 (2005), 1065-1185.[7] T.C. Hales, Dense Sphere Packings: A Blueprint for Formal Proofs, London Math. Society Lecture Note Series, Vol. 400, Cambridge University Press, Cambridge, 2012.[8] P. McMullen, Geometric Regular Polytopes, research monograph (in preparation). [9] P. McMullen and E. Schulte, Abstract Regular Polytopes, Cambridge University Press, Cambridge, 2002.[10] H.D. Mittelmann and F. Vallentin, High-accuracy semidefinite programming bounds for kissing numbers, Experimental Math. 19/2 (2010), 175-179.[11] D. Pellicer, A construction of higher rank chiral polytopes, Discrete Mathematics 310 (2010), 1222--1237.[12] D. Pellicer and E. Schulte, Regular polygonal complexes in space, I, II, Transactions Amer. Math. Society 362 (2010), 6679--6714 and 365 (2013), 2031--2061.